Probability Error Bounds for Approximation of Functions in Reproducing Kernel Hilbert Spaces. (3rd May 2021) - Aydın, Ata Deniz; Gheondea, Aurelian
Abstract : We find probability error bounds for approximations of functions f in a separable reproducing kernel Hilbert space H with reproducing kernel K on a base space X, firstly in terms of finite linear combinations of functions of type K x i and then in terms of the projection π x n on span K x i i = 1 n, for random sequences of points x = x i i in X . Given a probability measure P, letting P K be the measure defined by d P K x = K x, x d P x, x ∈ X, our approach is based on the nonexpansive operator L 2 X ; P K ∋ λ ↦ L P, K λ ≔ ∫ X λ x K x d P x ∈ H, where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by H P, that is the operator range of L P, K . Our main result establishes bounds, in terms of the operator L P, K, on the probability that the Hilbert space distance between an arbitrary function f in H and linear combinations of functions of type K x i, for x i i sampled independently from P, falls below a given threshold. For sequences of points x i i = 1 ∞ constituting a so-called uniqueness set, the orthogonal projections π x n to span K x i i = 1 n converge in the strong operator topology to the identity operator. We prove that, under the assumption that H P is dense in H, any sequence of points sampled independently from P yields a uniqueness set with probability 1. This result improves on previous error bounds in weaker norms, such as uniform or L p norms, which yield only convergence in probability and not almost certain convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space H 2 D are presented as well.